Study of the hydrodynamic of a flapping foil at moderate angle of attack O. Boiron a,b,⇑ , C. Guivier-Curien c , E. Bertrand b a Ecole Centrale Marseille, France b IRPHE, CNRS UMR7342, France c ISM, CNRS UMR6233, Marseille, France article info Article history: Received 17 June 2011 Received in revised form 22 October 2011 Accepted 19 January 2012 Available online 4 February 2012 Keywords: Foil ALE DPIV FSI abstract The mechanics of a rigid flapping foil animated by a combination of harmonic heave translation and pitch rotation is examined numerically and experimentally by Digital Particle Image Velocimetry (DPIV). The Arbitrary-Lagrangian–Eulerian (ALE) technique associated with a r-refinement grid adaption algorithm provides a good solution for studying the foil in a fixed reference frame while maintaining the grid quality over the whole simulation. Quantitative predictions were made, and showed very good agreement with the experimental data for a reduced frequency ranging in (0–0.6) and with the fixed values of Reynolds (4  10 3 ), heave amplitude (3c/4) and maximum angle of attack (20°). At low Strouhal numbers we observed a linear relationship between the thrust coefficient and the instantaneous angle of attack; for higher reduced frequency a non-linear behavior is observed, linked to added mass effects. Under the assumption of small angle of attack, these added mass effects correspond to a pitch stiffening and a heave damping. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Most fish and cetaceans move around their aquatic environment by flapping their tails. The high level of efficiency of flapping fins, along with their other useful properties such as their low-noise characteristics and manoeuvrability, has attracted some attention during the last few years. Several experimental, theoretical and numerical approaches have been used to assess the propulsive effi- ciency of flapping foils. Theodorsen [1] and Garrick [2] were the first authors to study how thrust is produced by plunging and/or pitch- ing airfoil. The inviscid thin airfoil method they employed using a conformal transformation approach can be used to predict the pro- pelling force in the whole frequency domain, with an efficiency of 50% in the case of infinitely rapid oscillations, and an efficiency of 100% was obtained asymptotically in that of infinitely slow oscilla- tions. Further improvements in the description of the problem were made during the years 1960–1980 using a non linear inviscid the- ory to account for the large amplitude effects. The role of the energy wasted in the wake was described by Ligh- till [3,4], and the role of the wake in general and more specifically, that of the vortex shedding occurring at the trailing edge, was established. The use of imaging methods, laser Doppler velocimetry (LDV) and Digital Particle Image Velocimetry (DPIV) made it possi- ble to describe qualitatively the vortex street which develops behind the trailing edge in detail. A jet-like wake associated with a reverse Kármán street is intrinsically associated with the genera- tion of thrust as it contributes to the production of momentum [5– 7]. Koochesfahani [8] showed the existence of an axial flow in the cores of the wake vortices using convection of dye. This axial flow is found to be under the influence of both the amplitude and the fre- quency of the foil motion. It is assumed by the authors to be related to the flow confinement by the channel walls. Triantafyllou et al. [9,10], using linear instability theory estab- lished that the optimum wake required to obtain high outputs occurs with an optimum set of non dimensional parameters, including in particular a non dimensional frequency (the Strouhal number) ranging between 0.25 and 0.35. Remarkably good agree- ment was found to exist between these results and a large body of data on fish(es) and cetaceans. Several authors performed fur- ther studies based on numerical simulations. The vortex lattice method or panel method was used for this purpose, assuming the presence/existence of an inviscid fluid. This problem has been addressed by Wang [11] and Guglielmini and Blondeaux [12] in the case of viscous flow, using a 2-D vorticity-stream function version of the Navier–Stokes equations in a non-inertial reference frame to model the airfoil. The results obtained with a wide range of param- eters showed that the production of thrust is accompanied by the generation of leading edge vortices which, combined with the trailing edge vorticity, give rise to the reverse Kármán street. Blondeaux et al. [13] confirmed numerically this complex behavior by performing a 3-D analysis of the wake behind a plunging/pitch- ing foil of finite span. They used the K ci criteria introduced by Zhou [14] to define the vortex structures. 0045-7930/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compfluid.2012.01.010 ⇑ Corresponding author at: Ecole Centrale Marseille, France. Tel.: +33 4 91 05 45 09; fax: +33 4 91 05 45 98. E-mail address: olivier.boiron@centrale-marseille.fr (O. Boiron). Computers & Fluids 59 (2012) 117–124 Contents lists available at SciVerse ScienceDirect Computers & Fluids journal homepage: www.elsevier.com/locate/compfluid